Problem: Find all values of $z$ such that $z^4 - 4z^2 + 3 = 0$.  Enter all the solutions, separated by commas.
Explanation: If we let $y=z^2$, then our equation becomes a simple quadratic equation:
$$y^2-4y+3=0.$$Indeed, this equation factors easily as $(y-3)(y-1)=0$, so either $y-3=0$ or $y-1=0$.

We now explore both possibilities.

If $y-3=0$, then $y=3$, so $z^2=3$, so $z=\pm\sqrt 3$.

If $y-1=0$, then $y=1$, so $z^2=1$, so $z=\pm 1$.

Thus we have four solutions to the original equation: $z=\boxed{-\sqrt{3},-1,1,\sqrt{3}}$.